Exothermic isospin-violating dark matter after SuperCDMS and CDEX

We show that exothermic isospin-violating dark matter (IVDM) can make the results of the latest CDMS-Si experiment consistent with recent null experiments, such as XENON10, XENON100, LUX, CDEX, and SuperCDMS, whereas for the CoGeNT experiment, a strong tension still persists. For CDMS-Si, separate exothermic dark matter or isospin-violating dark matter cannot fully ameliorate the tensions among these experiments; the tension disappears only if exothermic scattering is combined with an isospin-violating effect of f_n/f_p=-0.7. For such exothermic IVDM to exist, at least a new vector gauge boson (dark photon or dark Z') that connects SM quarks to Majorana-type DM particles is required.

We show that exothermic isospin-violating dark matter (IVDM) can make the results of the latest CDMS-Si experiment consistent with recent null experiments, such as XENON10, XENON100, LUX, CDEX, and SuperCDMS, whereas for the CoGeNT experiment, a strong tension still persists. For CDMS-Si, separate exothermic dark matter or isospin-violating dark matter cannot fully ameliorate the tensions among these experiments; the tension disappears only if exothermic scattering is combined with an isospin-violating effect of fn/fp = −0.7. For such exothermic IVDM to exist, at least a new vector gauge boson (dark photon or dark Z') that connects SM quarks to Majorana-type DM particles is required. expected to be reduced, and we shall see later that the DAMA result even shrinks to zero. Our strategy is first to apply exothermic DM to the SuperCDMS and CDEX results to confirm whether these more stringent experiments are consistent with the CDMS-Si result. If so, then exothermic DM becomes a unique type of DM that is consistent with all existing (except CoGeNT and DAMA) direct-detection experiments; if not, we will add in the IVDM effect and evaluate the results. As mentioned above, the IVDM model is already in tension with the results of LUX and CDMS-Si, and therefore, relying solely on IVDM to ameliorate the tensions among the different experiments is impossible; however, we can combine this mechanism with the exothermic DM model to further reduce these tensions.
For SuperCDMS, its latest result has recently been reported [25], in which the data obtained during 577 kg-days of exposure were analyzed for WIMPs of mass < 30 GeV/c 2 with a blinded signal region. Eleven events were observed once the analysis was complete. The authors set an upper limit on the spin-independent WIMP-nucleon cross section of 1.2 × 10 −42 cm 2 at 8 GeV/c 2 . In the meantime, CDEX already published its latest null results [26] for 53.9 kg-days of data.
To interpret the above experimental results in terms of inelastic scattering, we note that exothermic DM particles are those DM particles χ 1 of mass m 1 that inelastically down-scatter to DM particles χ 2 of mass m 2 from a nucleus N as follows: χ 1 + N → χ 2 + N . The requisite velocity to produce a nuclear recoil of energy E R is where µ is the reduced mass of the DM-nucleon system. Up-scattering (δ > 0) is more prevalent from heavy nuclei, whereas down-scattering (δ < 0) is more prevalent from light nuclei, where the energy of the recoiling nucleus is peaked near a scale that is proportional to the splitting between the dark matter states and is inversely proportional to the nuclear mass. Consequently, nuclear recoils caused by exothermic DM (δ < 0) are more visible in experiments with light nuclei and low thresholds. Given the lightness of Si with respect to Xe and Ge, down-scattering is one avenue for explaining the CDMS-Si data while remaining consistent with the null XENON, LUX, SuperCDMS, and CDEX searches. Figure 1 shows a plot of the elastic-scattering (corresponding to δ = 0) results from the CoGeNT, DAMA and CDMS-Si signal regions alongside the null results of XENON100, XENON10, LUX, SuperCDMS, and CDEX. For the DAMA experiment, it has been noted [58] that nuclei recoiling along the characteristic axes or planes of the crystal structure may travel large distances without colliding with other nuclei. This means that recoils that undergo such ion channeling have quenching factors of Q T ≈ 1. We consider the cases both with and without this ion-channeling effect. The null experiments of XENON100, LUX and SuperCDMS are in strong tension with the CoGeNT, DAMA (both with and without the ion-channeling effect), and CDMS-Si results. In Fig. 2, we consider the results for the inelastic scattering of exothermic DM (corresponding to δ < 0) from the CoGeNT and CDMS-Si signal regions along with the null results from XENON100, XENON10, LUX, SuperCDMS, and CDEX. The left panel corresponds to δ = −50 keV, and the right panel corresponds to δ = −200 keV. We see that for δ = −50 keV, the situation is slightly improved, whereas for δ = −200 keV, the situation is much improved. The exception is CoGeNT, which is, as expected, still in tension with all null experiments; XENON10, XENON100 (lying outside the plot area to the right) and CDEX are already consistent with CDMS-Si. LUX covers almost the entire CDMS-Si contour for δ = −50 keV but covers only a small portion for δ = −200 keV. Only SuperCDMS still fully covers the CDMS-Si contour and is strongly in tension with the CDMS-Si result. One observation is that the signal region of CoGeNT becomes significantly larger than that for CDMS-Si at δ = −200 keV. This behavior arises from the difficulty in fitting the data from the multi-events to a relatively large DM mass splitting [57]. The χ 2 min of this fitting is significantly larger than that for the elastic fitting. Because δ = −200 keV is already approaching the lower limit on the allowed mass difference for exothermic DM [54], the results of Fig. 2 indicate that exothermic DM alone, even when an extreme δ value is used and the CoGeNT result is ignored, is still not sufficient to accommodate both the SuperCDMS and CDMS-Si results. For DAMA, note that when the inelastic scattering of DM is considered, the area of the low-mass signal region from the DAMA experiment shown in Fig. 1 reduces as the mass splitting |δ| grows. This effect can also be observed in Fig. 1 of [52]. In our analysis with δ = −50 keV and δ = −200 keV, the signal region of DAMA for low masses (masses comparable to the signal regions of CDMS-Si and CoGeNT) completely disappears, or the DAMA result shrinks to zero. For this reason, in the following, as long as we are discussing exothermic DM with δ = −50 keV or δ = −200 keV, we shall no longer consider the DAMA experiment. Furthermore, the inelastic-scattering DM does not fit the DAMA data well even for larger masses (m χ > 30GeV); the χ 2 min /d.o.f is approximately 35/34 for δ = −200 keV, whereas χ 2 min /d.o.f =27.8/34 for δ = 0. Next, we include an isospin-violating effect. The general low-energy differential cross section is [50] dσ where Z is the atomic number of the target nucleus; A is its mass number; f p and f n are constants that represent the relative coupling strengths to protons and neutrons, respectively; and F (q 2 ) a form factor that depends on the momentum transfer to the nucleus, q 2 = 2m N E R . σ el is the elastic limit of the above cross section, which is reached when the splitting is much less than the kinetic energy of the collision. The differential scattering rate of dark matter per unit recoil energy E R is given by where v min , which is determined using Eq. (1), is the minimum velocity required to produce a recoil of energy E R ; N T is the number of target nuclei; n χ is the local number density of the dark matter; and f (v) is the distribution of DM velocities relative to the target. With N T m N = m detector and ρ χ = n χ m χ , the differential recoil rate per unit detector mass can be written as where ρ χ = 0.3 GeV/c 3 is the local DM density. Details of the DM velocity distribution are included via the mean inverse speed η(E, t), where f (v) at any given time of the year is determined by the velocity of the Earth through the halo and by the distribution of DM velocities within the halo itself, here assumed to be of the form We have assumed a Maxwell-Boltzmann distribution for the DM halo velocities with a mean of v 0 = 220km/s and a sharp cutoff (i.e., the galactic escape velocity) at v esc = 544km/s. N 0 is chosen to normalize the probability distribution to one. Because the Earth is moving with a velocity v E = 220km/s, η(E, t) can be written as [2] η(E, t) = where For the annual modulation, the count rate generally has an approximate time dependence as follows: where t c is the time of year at which v obs (t) is at its maximum, S 0 (E R ) is the average differential recoil rate over a year, and S m (E R ) is referred to as the modulation amplitude. For the standard halo model, where v orb = 30km/s and cos γ = 0.51. Finally, to consider isospin-violating scattering from dark matter, different mass numbers will yield different differential recoil rates. The event rate is given by where the sum is over the isotopes A i with fractional number abundances r i [39]. Using these formulae, and with a ratio of f n /f p ≈ −0.7, we performed the relevant calculations, and in Fig. 3, we plot the elastic-scattering (corresponding to δ = 0) IVDM results of the CoGeNT, DAMA (both with and without the ion-channeling effect) and CDMS-Si signal regions, alongside the null results of XENON100, XENON10, LUX, SuperCDMS, and CDEX. Through comparison with Fig. 1, we find that the IVDM model does slightly reduce the tensions, but the null experiments LUX and SuperCDMS are essentially still in tension with the CoGeNT, DAMA and CDMS-Si result. In particular, we recover the previously mentioned result that LUX and CDMS-Si are in tension for IVDM [45][46][47].
We continue by considering the inelastic-scattering effects. The underlying model for inelastic scattering is typically constructed with a vector particle-dark photon (or dark Z') mixing kinetically with an SM U(1) gauge boson and coupling to the two different DM particles, χ 1 and χ 2 [49,57]; here, to ensure that the coupling of the DM particles to the dark photon is strictly off-diagonal in the mass basis, the DM particles must be Majorana states because there then exists no vector current for a single Majorana particle. In this scenario, elastic scattering between DM and nucleons can occur happen at second order (right panel of Fig. 4) and is thus suppressed, whereas inelastic scattering can occur at first order (left panel of Fig. 4) and thus plays the leading role in direct-detection experiments. If, furthermore, the kinetic energy is smaller than the mass splitting of the DM, then up-scattering on nucleons is kinetically prohibited, and we are left with the exothermic scattering of the DM. To further account for the large isospin-violating effect, the conventional Higgs portal scheme of a scalar field mixing with the SM Higgs to communicate between the SM and DM sectors causes no significant isospin violation [59] because only a very small percentage of the nucleon constituents are related to the current quark mass and thus connected to the Higgs field. We then must exploit vector instead of scalar particles to connect the dark world with SM particles 2 . For such a model with a single messenger, the isospin-violating effect depends on the choice of SM U(1) with which the new vector boson mixes. For example, if, as usual, we take U(1) to be the SM hypercharge U(1) Y [57], because the proton and neutron have the same hypercharge, we then expect the plot for the left diagram of Fig. 4 to be the same for both neutrons and protons, leading to f n = f p , i.e., there is no isospin violation. If, instead, we consider that in the low-energy region, the Z-boson component of U(1) Y decouples, then effectively, only the electromagnetic part will contribute, and we can then take U(1) to be the SM electromagnetic U(1) em [49,54]; because the neutron is neutral and the proton is charged, we then expect the same plot to be zero for neutrons, resulting in f n = 0 and f p = 0, i.e., we have isospin violation. In Fig. 5, we plot the f n = 0 IVDM exothermic DM result for the CoGeNT and CDMS-Si signal regions along with the null results of XENON10, LUX, SuperCDMS, and CDEX (XENON100 lies to the right, outside the plot area).
Comparison of the plots of Fig. 5 and the right panel of Fig. 2 reveals only a very few changes. In particular, the SuperCDMS result is only marginally in tension with the CDMS-Si result. This is because the maximum suppression values of f n /f p are −0.785 (for Ge), −0.697 (for Xe), and −0.992 (for Si), and a detailed computation shows that if we take f n /f p = −0.7, then the energy spectra of Ge and Xe relative to Si are suppressed by approximately 90% and 95%, respectively; if we set f n = 0, then the suppression of Ge and Xe relative to the Si energy spectra is reduced by 20%. Hence, f n = 0 offers an insufficient isospin-violating effect, and we must increase its strength by setting f n /f p = −0.7. In the literature, the first discussion of vector boson exchange leading to f n /f p = −0.7 was presented in Ref. [61], and in that discussion, the key roles were played by three factors: the conventional kinetic mixing and the mass mixing between SM U (1) and the dark photon or Z' as well as the coupling of the dark photon to SM quarks. Although the original model presented in Ref. [61] does not include the inelastic-scattering effect, we can modify the model by adding a Majorana mass term to the DM fields, which will yield exactly an off-diagonal dark-photon coupling to the DM fermions, and this improvement does not change the value of f n /f p . To be more explicit, we write a Lagrangian for our proposed schematic model as follows: where the extra U (1) X is assumed to be broken and the corresponding vector boson mass is m Z ′ . We denote the fields in the interaction basis by (B, W 3 , X) and in the mass-eigenstate basis by (A, Z, Z ′ ), and we define Z ≡ c W W 3 −s W B, where s W (c W ) is the sine (cosine) of the Weinberg angle. χ is the fermionic DM field with Dirac mass M and Majorana mass δ ≪ M . For this Lagrangian, the discussions of Ref. [61] demonstrate that there exist suitable parameter spaces (ǫ, δm 2 , f V f ) to account for f n /f p = −0.7, as described in greater detail below.
• For the dark Z' scenario, in which the SM fields are uncharged under the extra U (1) X group and, thus, f V f = 0, Fig. 2 of Ref. [61] shows that the ratio f n /f p ∼ 0.7 with m Z ′ = 4 GeV can be achieved by adjusting the remaining two parameters ǫ and δm 2 appropriately. The figure shows that for ǫ ≈ δm 2 /m 2 Z and ǫ ≪ 1, we have f n /f p ≈ 1/3s W ≈ −0.7.
• For the baryonic Z' scenario, the SM is charged under the U (1) X group, whereas the leptons are uncharged under U (1) X and U (1) Figure. 3 of Ref. [61] shows that the ratio f n /f p ∼ 0.7 can be achieved by adjusting two of the three parameters; the left panel illustrates the variation of ǫ and f V q with δm 2 = 0, and the right panel illustrates the variation of ǫ and δm 2 with f V q ≈ 10 −5 . The figure shows that to obtain f n /f p ≈ −0.7, f V q must be more than an order of magnitude smaller than ǫ. Suppose that ǫ in Ref. [61] is constrained to be on the order of 10 −2 or smaller, such that f V q ≤ 10 −3 . The requisite smallness of f V q may be achieved by coupling Z ′ only to the second-and third-generation quarks, and this relaxes the restriction that the additional U (1) X must be baryonic, thereby allowing for couplings to leptons to facilitate the construction of an anomaly-free model. By contrast, diagonalizing the DM mass matrix leads to mass eigenstates χ 1,2 of masses M 1,2 = m χ ∓ δ and an off-diagonal gauge interaction, which leads to the DM scattering picture previously considered in Fig. 4.
Ref. [62] presents similar discussions with two additional important extensions: first, noting the possibility of applying the model to inelastic scattering, and second, proving that a combination with the conventional Higgs mediator will help to achieve the desired isospin violation. These extensions are also investigated in Ref. [59], and the combination of the dark photon and the conventional Higgs mediator is further generalized to the combination of the dark photon and another new light vector gauge boson. Using our schematic model (12), especially the parameter range represented in Fig. 2 and Fig. 3 of Ref. [61], in addition to these other possible underlying exothermic IVDM models that give rise to an expected value of f n /f p = −0.7, we plot the IVDM exothermic DM result for the CoGeNT and CDMS-Si signal regions along with the null results of XENON100, XENON10, LUX, SuperCDMS, and CDEX (see Fig. 6). The left plot corresponds to δ = −50 keV, and the right plot corresponds to δ = −200 keV. Apart from the strong tension remaining between CoGeNT and the null experiments, we see that even for δ = −50 keV, CDMS-Si is already consistent with most of the null experiments, although LUX cuts through half of the contour. For δ = −200 keV, the tensions between CDMS-Si and the null experiments are over-relaxed. Therefore, with the assistance of isospin-violating effects from the dark photon or Z', we can readily make CDMS-Si consistent with all current null experiments, even without invoking the extreme case of exothermic DM with δ = −200 keV. This leaves open a region in the parameter space for exothermic DM to fit other current and future DM detection experiments.
It should be noted that there are several other possible methods of reconciling the tensions among various directdetection experiments. The first is to interpret the possible signals appearing in DAMA, CoGeNT, and CDMS-Si not as DM signals but as some atmospherically produced neutral particle with a relatively large magnetic dipole moment [63], as such particles can mimic DM signals. A very definite flux could explain the signals observed in DAMA/LIBRA, CDMS-Si, and CoGeNT that are consistent with the bounds from XENON100 and CDMS-II. In this scenario, the key is that the recoil energy of the assumed particle must lie some specific energy range that is above the thresholds of DAMA/LIBRA, CDMS-Si, and CoGeNT but below those of XENON100 and CDMS-II. If we further consider the latest results of SuperCDMS and CDEX, then this recoil energy must lie above the thresholds of these two experiments and therefore is expected to produce signals in these detectors. This has is not occurred, hence implying that this alternative interpretation is not favored by the latest SuperCDMS and CDEX null results.
The second possibility is to invoke composite DM, wherein stable particles of charge 2 bind with primordial helium to form O-helium "atoms" (OHe), representing a specific warmer-than-cold nuclear-interacting form of dark matter [64]. Because it slows down in terrestrial matter, OHe is elusive in direct methods of underground DM detection such as those used in the CDMS experiment, but its reactions with nuclei can lead to annual variations in the energy released in the energy range of 2 − 6 keV such as those observed in the DAMA/NaI and DAMA/LIBRA experiments. However, this class of solution cannot explain the unmodulated signals in experiments such as CoGeNT and CDMS-Si and therefore is not favored by these experiments.
Finally, for completeness, we will list for each experiment some of the details of the computations used to obtain all the above plots (except Figs. 4): (i) CDMS-Si: We used the acceptance from [11] and a total exposure of 140.2 kg-days, assuming zero background.
We considered an energy interval of [7,100] keV and binned the data in 2 keV intervals as in [47]. The three candidate events appeared in the first three bins. To find the best-fit regions, we obtained the extended loglikelihood function and simply plotted constant values of the likelihood that it would correspond to the 68% and 95% CL regions under the assumption that the likelihood distribution is Gaussian.
(ii) CoGeNT: We used the data and flat background shown in Fig. 23 of [7], which has been corrected for efficiency (i.e., bin counts have been scaled to reflect the numbers of events expected based on those observed and the deduced efficiency). We performed a χ 2 scan over a cross section using the DM mass and background from the data of Ref. [7]. The curves for the region of interest correspond to the 90% C.L. regions. The energy resolution below 10 keV was taken to be that reported by CoGeNT, namely, σ 2 = σ 2 n + 2.35 2 EηF , where σ n = 69.4 eV is the intrinsic electronic noise, E is the energy in eV, η = 2.96 eV is the average energy required to create an electron-hole pair in Ge at approximately 80 K, and F = 0.29 is the Fano factor. The number of expected events in a given range was taken to be [45] where b is the flat, floating background and 2res(E 1 , (iii) DAMA: The average amplitude over the energy interval [E 1 , where c T is the mass fraction of the target and Q T is the quenching factor for the target, which we take to be Q N a = 0.3 and Q I = 0.09. To account for the ion-channeling effect, we take the channeling fraction to be as in Ref. [58]. The measured energy will be normally distributed with a standard deviation of We used the data presented in Fig. 6 of [3]. We calculated χ 2 using all 36 bins corresponding to energies from 2 keV to 20 keV. The 95% C.L. contours of the region of interest satisfy χ 2 = χ 2 min + CDF −1 (ChiSq [2], C.L.). (iv) XENON10: We simply adopted the collaboration's parameterization from Fig. 1 of [16], assuming a sharp cutoff to zero at a nuclear recoil energy of 1.4 keV. The signal region is from 5 to 35 electrons, corresponding to nuclear recoils of 1.4 keV to 10 keV. A limit of 90% C.L. was obtained using the p max method [65] and the 23 highlighted S2 event signals from Fig. 2 of [16].
(v) XENON100: We used the mean ν(E) characterized in [66]. For the scintillation efficiency L ef f , we used the efficiency used in XENON100's 225-live-day analysis [19] obtained from Fig. 1 of [17], which included a linear extrapolation to 0 for E below 3 keV. The response of the detector was modeled as a Gaussian distribution with a mean of n and a variance of √ nσ P MT , where σ P MT = 0.5PE [66]. The Gaussian smearing also included a photoelectron-dependent acceptance, which we parameterized based on Fig. 1 of [19]. To obtain the total rate, we summed the differential rate over the signal region, which corresponds to S1 ∈ (3, 30)PE for the analysis presented in [19], and used a total exposure of 225×34 kg-days [19]. We then used Poisson statistics to obtain a 90% C.L. upper limit, where two events were observed, as shown in Fig. 2 of [19].
(vi) LUX: The experimental design of LUX is quite similar to that of XENON100 [57]. Both experiments use a combination of scintillation (S1) and ionization signals (S2) to effectively reject background. Following [66], we computed the number of signal events as follows: Gauss(S1|n, where Ex. denotes the experimental exposure, ǫ(E R ) is the S1 efficiency, and σ P MT = 0.37 PE accounts for the PMT resolution. For the LUX analysis, S1 lower = 2 and S1 upper = 30. The expected number of photoelectrons ν(E R ) is where L ef f is the energy-dependent scintillation efficiency of liquid xenon, L y is the light yield, and S n and S e are the nuclear-and electron-recoil quenching factors, respectively, that arise from the applied electric field. We used the energy-dependent absolute light yield, L ef f (E R ) Sn Se L y , from slide 25 of [67], with a hard cutoff below 3 keV. Finally, for the DM detection efficiency, we used the efficiency calculated after threshold cuts from Fig. 9 of [24]. We computed 90% CL limits using Poisson statistics with no events detected.
(vii) SuperCDMS: For the efficiency reported in Fig. 1 of [25], we used the 577-kg-day data from [25]. To obtain the 90% C.L. limits, we used Poisson statistics with 11 candidate events detected, which are listed in Table 1 of [25], and zero background was assumed.
(viii) CDEX: We assumed perfect efficiency and used the 53.9-kg-day data from the residual spectrum presented in Fig. 3(b) of [26]. A flat background was assumed, as given by the minimum χ 2 method. The quenching factor of a recoiling Ge nucleus was obtained from the TRIM program as in [68]. To obtain the 90% C.L. limits, the binned Poisson method [2] with bins of 0.1 keVee was used.
To summarize, we find that exothermic DM alone is not sufficient to fully resolve the tensions between CDMS-Si and the null experiments. However, if some underlying interaction allows isospin-violating effects to be incorporated into exothermic DM models, then, with the aid of the strongest setting f n /f p = −0.7, exothermic IVDM can make the CDMS-Si result consistent with the results of all the latest null experiments, except the CoGeNT experiment. Meanwhile, for exothermic IVDM to exist, at least a new vector gauge boson (dark photon or dark Z') that connects SM quarks with Majorana-type DM particles is required.